Course Name: Eight Grade Math Unit # 2 Unit Title: Thinking with Mathematical Models

Course Name: Eight Grade Math Unit # 2 Unit Title: Thinking With Mathematical Models

CORE CONTENT

Enduring understanding (Big Idea): Students will model relationships with graphs and equations. The will use models to analyze situations and solve problems. Students will understand that solving multi-step equations connect through tables and graphs and students will understand the difference between a linear and a nonlinear function.
Essential Questions: What are the key variables in this situation? If there is a pattern relating variables, is it strong enough to allow me to make a predictions? What is the pattern relating the variables? What kind of equation will express the relationship? How can I use the equation to answer questions about the relationship?

BY THE END OF THIS UNIT:

Unit Plans / Investigation / Suggested ACE Questions
Standards 8.F.2; 8.F.3; 8.F.5; 8.SP.1
Investigation 1
Exploring Data Patterns / 1.1 Bridge Thickness and Strength
1.2 Bridge Length and Strength
1.3 Custom Construction Parts
Math Reflections
Common Core Mathematical Practices / 1.1: ACE
1.2: ACE
1.3: ACE
Standards 8.EE.5; 8.EE.7b; 8.EE.8a; 8.EE.8c; 8.F.3; 8.F.4; 8.SP.2
Investigation 2
Linear Models and Equations / 2.1 Modeling Linear Data Patterns
2.2 Up and Down the Staircase
2.3 Tree Top Fun
2.4 Boat Rental Business
2.5 Amusement Park or Movies
Math Reflections
Common Core Mathematical Practices / 2.1: ACE
2.2: ACE
2.3: ACE
2.4: ACE
2.5: ACE
Standards 8.EE.5; 8.F.3; 8.F.5; 8.SP.1
Investigation 3
Inverse Variation / 3.1 Rectangles With Fixed Area
3.2 Distance, Speed, and Time
3.3 Planning a Field Trip
3.4 Modeling Data Patterns
Math Reflections
Common Core Mathematical Practices / 3.1: ACE
3.2: ACE
3.3: ACE
3.4: ACE
Standards 8.SP.1; 8.SP.2; 8.SP.3
Investigation 5
Variability and Associations in Categorical Data / 5.1 Wood or Steel? That’s the Questions
5.2 Politics of Girls and Boys
5.3 After-School Jobs and Homework
Math Reflections
Common Core Mathematical Practices / 5.1: ACE
5.2: ACE
5.3: ACE

*The ACE will be completed when the teacher resources arrive.

CORE CONTENT

Cluster Title: Define, evaluate, and compare functions.
Standard 8.F.2: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Concepts and Skills to Master
● Compare two linear functions each represented a different way and describe similarities and differences in slopes, y-intercepts, and values.

SUPPORTS FOR TEACHERS

Critical Background Knowledge
● Determine slopes and y-intercepts.
Academic Vocabulary
slope, intercept, rate of change, function, linear, non-linear
Suggested Instructional Strategies
●  Given one representation of a function, create the others.
●  Put students in small groups. Give groups scenarios and ask each group to create a different representation of the scenario (table, equation, graph).
●  Identify attributes (slope, y-intercept, values) of a function in its equation, graph, or a table. / Resources
Textbook Correlation
●  Thinking With Mathematical Models
○  Investigation 1
●  Say It With Symbols
○  Investigation 2
Helpful Websites / Resources
○  Many links to appropriate resources connected to 8.F.2 - http://ccssmath.org/?page_id=715
○  8th Grade Common Core Math Wiki
○  Internet 4 Classroom Resources
○  Real World Situations - Comparing Functions
○  MARS Concept AssessmentTaskLessons (MS):
A05: Baseball Jerseys; A17: Linear Graphs, A19: Meal Out
○  MARS Concept Formative AssessmentLessons (MS):
Modeling Situations With Linear
○  CMP2 Resources
○  Texas Instrument 8.F.2 Lessons:http://education.ti.com/calculators/downloads/US/Activities/Search/Standards
○  YouTube Video: Algebra: Graphing Lines 1
○  YouTube Video: Graphing Linear Equations
Sample Assessment Tasks
Skill-based Task
Is y=2(x+5) the same as the function described as “twice a quantity plus 5”? / Problem Task
Billy argues that the equation y=4x+5 is equivalent to the equation of the line that goes through (2,6) and (3,10). How did he arrive at this conclusion? Is he correct? Justify your answer.

CORE CONTENT

Cluster Title: Define, evaluate, and compare functions.
Standard 8.F.3: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
Concepts and Skills to Master
●  Distinguish between linear and non-linear functions given their algebraic expression, a table, or a graph.
●  Recognize that functions written in the form y = mx + b are linear and that every linear function can be written in the form y= mx + b

SUPPORTS FOR TEACHERS

Critical Background Knowledge
●  Generate and plot ordered pairs from an equation.
●  Understand linear slope as a constant rate of change.
Academic Vocabulary
collinear, linear, nonlinear
Suggested Instructional Strategies
●  Examine constant and non-constant rates of change in tables of values.
●  Explore growing patterns generated from a variety of contexts to explore linear and nonlinear relationships. / Resources
Textbook Correlation
●  Thinking With Mathematical Models
○  Investigation 2
Helpful Websites / Resources
●  Many links to appropriate resources connected to 8.F.3 - http://ccssmath.org/?page_id=717
●  Grapher
●  Line Plots
●  MARS Concept AssessmentTaskLessons (MS):
○  A05: Baseball Jerseys; A17: Linear Graphs, A19: Meal Out
●  CMP2 Resources
●  MARS Concept Formative AssessmentLessons (MS):
○  Modeling Situations With Linear
●  Texas Instrument 8.F.3 Lessons: http://education.ti.com/calculators/downloads/US/Activities/Search/Standards
Sample Assessment Tasks
Skill-based Task
Determine which of the following equations are linear:
a) y = 3x + 1 b) y = x c) 2x – y = 5
d) y = 2x e) 2y + 5x² = 0 / Problem Task
Hermione argues that the table below represents a linear function. Is she correct? How do you know?
x / 10 / 8 / 6 / 2 / 0
y / -1 / 3 / 7 / 15 / 19

CORE CONTENT

Support for Teachers

Cluster Title: Use functions to model relationships between quantities.
Standard 8.F.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x , y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Concepts and Skills to Master
○  Determine and interpret the initial value and rate of change given two points, a graph, and a table of values, a geometric representation, or a story problem (verbal description) of a linear relationship.
○  Write the equation of a line given two points, a graph, a table of values, a geometric representation, or a story problem (verbal description) of a linear relationship.
Critical Background Knowledge
● Understand the meaning of slope and y-intercept; Write an equation as y = mx + b given a graph.
Academic Vocabulary
Linear relationship, y-intercept, slope
Suggested Instructional Strategies
●  Use a real-world application to generate a table of values.
●  Use the table to construct a function that models the relationship.
●  Connect to other standards in the Expressions and Equations Domain. / Resources
Textbook Correlation
●  Thinking With Mathematical Models
○  Investigations 1 and 2
Helpful Websites / Resources
●  Many links to appropriate resources connected to 8.F.4 - http://ccssmath.org/?page_id=719
●  YouTube Video:Algebra - Find the Equation of a Line Given Two Points Intuitive Math Help
●  CMP2 Resources
●  MARS Concept AssessmentTaskLessons (MS):
●  A05: Baseball Jerseys; A17: Linear Graphs, A19: Meal Out
●  MARS Concept Formative AssessmentLessons (MS):
●  Modeling Situations With Linear
●  Texas Instrument 8.F.4 Lessons: http://education.ti.com/calculators/downloads/US/Activities/Search/Standards
●  Video Streaming
●  Baseball Cards
●  Chicken and Steak, variation 1
Sample Assessment Tasks
Skill-based Task
●  The student council is planning a ski trip to Sundance. There is a $220 deposit for the lodge and the tickets will cost $70 per student. Construct a function, build a table, and graph the data showing how much it will cost for the students’ trip.
●  Find the equation of the line that goes through (3,5) and (-5,7). / Problem Task
Michael says that the equation of a line that passes through the points (2,4) and (-4 , -6) is y = -2x + 2. Is he correct? Explain why or why not.
Wally created the table below for a function he knows to be linear. He thinks something must be wrong with his table because he can’t find the original function from the table. Find the error and the original function. Explain your strategy for finding the error.
3.2 6.4 9.6 12.8 16 19.2 22.4 25.6
17.8 30.6 43.4 56.2 66 81.8 94.6 107.4

CORE CONTENT

Cluster Title: Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers.
Standard 7.NS.1c: Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real world contexts.
Concepts and Skills to Master
●  Ability to apply and extend previous understanding to represent addition and subtraction problems of rational numbers with a horizontal or vertical number line.
●  Ability to understand the relationship between a positive number and its opposite
●  Ability to write mathematics sentences to show relationships
●  Ability to use appropriate notations to indicate positive and negative numbers

SUPPORTS FOR TEACHERS

Critical Background Knowledge
Conceptual
●  Understand how to model addition and subtraction of integers using distance/direction on a number line and using chips.
●  Understand that the Commutative Property holds for addition of rational numbers.
●  Understand and use the relationship between addition and subtraction to simplify computation by changing subtraction problems to addition and vice versa.
●  Understand and use the relationship between addition and subtraction found in fact families.
Procedural
●  Develop algorithms for adding and subtracting integers.
●  Recognize and solve problems involving addition and subtraction of integers.
●  Use the Distributive Property to solve problems.
●  Solve simple equations with missing facts by using fact families
Academic Vocabulary
Additive inverse, rational numbers
Suggested Instructional Strategies
●  To introduce students to adding integers, discuss examples of saving and spending.
●  Start the lesson by showing students the addition of two positive numbers. Then show that addition of two negative numbers follow a similar pattern. Focus your presentation on integers with different signs. Use the number line to model.
●  To introduce students to subtracting integers, ask them to explain how they would subtract a greater number from a lesser number.
●  To introduce subtraction of integers, relate negative integers to borrowing money to buy something you do not have enough money for. / Resources
Textbook Correlation
●  Accentuate the Negative
○  Investigations 1, 2, and 4
Helpful Websites / Resources
●  Many links to appropriate resources connected to 7.NS.1c - http://ccssmath.org/?page_id=616
●  MARS Task:
●  A11: Division
●  E03: A Day Out
●  E11: Taxi Cabs
●  Texas Instruments Lessons:
●  Adding Integers – A Modeling Approach
●  Adding Integers Exploration
●  Getting Negative
●  Integer Subtraction – What’s the Difference?
●  Integers
●  Number Line Activity – Adding Integers
●  http://mathstar.lacoe.edu/lessonlinks/integers/integers_adding_main.html
●  http://mathstar.lacoe.edu/lessonlinks/integers/integers_subtracting_main.html
●  http://www.uen.org/Lessonplan/preview?LPid=23406
Sample Assessment Tasks
Skill-based Task
Morgan has $4 and she needs to pay a friend $3. How much will Morgan have after paying her friend? / Problem Task
The Ravens started their possession on the 20 yard line. On 1st down, they gained 4 yards. On 2nd down they lost 9 yards. What is their total yardage so far?

CORE CONTENT

Cluster Title: Apply and extend previous understandings of operations with fractions to add, subtract, multiply and divide rational numbers.
Standard 7.NS.1d: Apply properties of operations as strategies to add and subtract rational numbers.
Concepts and Skills to Master:
●  Ability to apply and extend previous understanding to represent addition and subtraction problems of rational numbers with a horizontal or vertical number line.
●  Ability to understand the relationship between a positive number and its opposite
●  Ability to write mathematics sentences to show relationships
●  Ability to use appropriate notations to indicate positive and negative numbers

SUPPORTS FOR TEACHERS

Critical Background Knowledge
Conceptual
●  Understand how to model addition and subtraction of integers using distance/direction on a number line and using chips.
●  Understand that the Commutative Property holds for addition of rational numbers.
●  Understand and use the relationship between addition and subtraction to simplify computation by changing subtraction problems to addition and vice versa.
●  Understand and use the relationship between addition and subtraction found in fact families.
Procedural
●  Develop algorithms for adding and subtracting integers.
●  Recognize and solve problems involving addition and subtraction of integers.
●  Use the Distributive Property to solve problems.
●  Solve simple equations with missing facts by using fact families
Academic Vocabulary: Additive inverse, rational numbers
Suggested Instructional Strategies
●  To introduce students to adding integers, discuss examples of saving and spending.
●  Start the lesson by showing students the addition of two positive numbers. Then show that addition of two negative numbers follow a similar pattern. Focus your presentation on integers with different signs. Use the number line to model.
●  To introduce students to subtracting integers, ask them to explain how they would subtract a greater number from a lesser number.
●  To introduce subtraction of integers, relate negative integers to borrowing money to buy something you do not have enough money for. / Resources
Textbook Correlation
●  Accentuate the Negative
○  Investigations 1, 2, and 4
Helpful Websites / Resources
●  Many links to appropriate resources connected to 7.NS.1d - http://ccssmath.org/?page_id=618
●  MARS Task:
●  A11: Division
●  E03: A Day Out
●  E11: Taxi Cabs
●  Texas Instruments Lessons:
●  Adding Integers – A Modeling Approach
●  Adding Integers Exploration
●  Getting Negative
●  Integer Subtraction – What’s the Difference?
●  Integers
●  Number Line Activity – Adding Integers
●  http://mathstar.lacoe.edu/lessonlinks/integers/integers_adding_main.html
●  http://mathstar.lacoe.edu/lessonlinks/integers/integers_subtracting_main.html
●  http://www.uen.org/Lessonplan/preview?LPid=23406
Sample Assessment Tasks
Skill-based Task
1.  On 3rd down of the same possession they lost 7 more yards. What is their total yardage now? / Problem Task
Introduce the situation: Chris, Rob, Amy and Melissa are arguing over which day of their skiing trip the temperature dropped the most. Below are the scenarios for each of the 4 days of their trip and who picked each day as the greatest drop in temperature.
a.  Rob: On Friday, when they hit the slopes in the morning the temperature was 12.5°F. When they finally called it quits that evening, the temperature had dropped to -3.5°F. (16°F)
b.  Melissa: On Saturday when they got to the ice skating rink in the morning, the temperature was -11.5°. When they were having hot cocoa that evening the temperature had dropped to -17.25°F. (5.75°F)
c.  Amy: On Sunday morning when they left the hotel to go snowboarding the temperature outside was 5.75°F. When they stopped for dinner that evening, the temperature had dropped to -7.5°F. (13.25°F)
d.  Chris: On Monday morning as they were hitting the slopes on the last day of their trip, the temperature was -2.25°F. When they finally left the slopes that evening to go home and pack, the temperature had dropped to -21.5°F. (19.25°F)

CORE CONTENT